I wouldn't want to deal with health issues right out of the gate (like I wouldn't buy a dysplastic pup for example) but things that come up, torn ACL, allergies, eh, it sucks but it happens.
Mostly what I'm concerned with is proper temperament. I wouldn't want an HA or shy APBT, I wouldn't want a silly friendly Rottweiler, I wouldn't want a hunting or herding dog that couldn't perform drive-wise to the breed standard. Things of that nature.
My hound mutt has stolen my heart and I would like to keep a beagle or beagle mix forever alongside my purebreds. For my little muttley sidekicks, I only require a kind, biddable temperament.